Theorem cusgrres 29764

Description: Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)

Hypotheses

Ref	Expression
cusgrres.v	⊢ 𝑉 = (Vtx‘𝐺)
cusgrres.e	⊢ 𝐸 = (Edg‘𝐺)
cusgrres.f	⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
cusgrres.s	⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩

Assertion

Ref	Expression
cusgrres	⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ ComplUSGraph)

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉

Allowed substitution hints:   𝑆(𝑒)   𝐹(𝑒)

Proof of Theorem cusgrres

Dummy variables 𝑛 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cusgrusgr 29735	. . 3 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)
2		cusgrres.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3		cusgrres.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
4		cusgrres.f	. . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
5		cusgrres.s	. . . 4 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
6	2, 3, 4, 5	usgrres1 29631	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph)
7	1, 6	sylan 591	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph)
8		iscusgr 29734	. . . 4 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
9		usgrupgr 29501	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
10	9	adantr 485	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) → 𝐺 ∈ UPGraph)
11	10	anim1i 626	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) → (𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉))
12	11	anim1i 626	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})))
13	2	iscplgr 29731	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺)))
14		eldifi 4084	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ 𝑉)
15	14	ad2antll 741	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USGraph ∧ (𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁}))) → 𝑣 ∈ 𝑉)
16		eleq1w 2844	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑣 → (𝑛 ∈ (UnivVtx‘𝐺) ↔ 𝑣 ∈ (UnivVtx‘𝐺)))
17	16	rspcv 3576	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 → (∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝐺)))
18	15, 17	syl 18	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ USGraph ∧ (𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁}))) → (∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝐺)))
19	18	ex 417	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → ((𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝐺))))
20	19	com23 87	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺) → ((𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺))))
21	13, 20	sylbid 243	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph → ((𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺))))
22	21	imp 411	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) → ((𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺)))
23	22	impl 460	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺))
24	2, 3, 4, 5	uvtxupgrres 29724	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (𝑣 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝑆)))
25	12, 23, 24	sylc 66	. . . . 5 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝑆))
26	25	ralrimiva 3155	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆))
27	8, 26	sylanb 592	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆))
28		opex 5445	. . . . 5 ⊢ ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
29	5, 28	eqeltri 2857	. . . 4 ⊢ 𝑆 ∈ V
30	2, 3, 4, 5	upgrres1lem2 29627	. . . . . 6 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
31	30	eqcomi 2770	. . . . 5 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
32	31	iscplgr 29731	. . . 4 ⊢ (𝑆 ∈ V → (𝑆 ∈ ComplGraph ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆)))
33	29, 32	mp1i 14	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ ComplGraph ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆)))
34	27, 33	mpbird 260	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ ComplGraph)
35		iscusgr 29734	. 2 ⊢ (𝑆 ∈ ComplUSGraph ↔ (𝑆 ∈ USGraph ∧ 𝑆 ∈ ComplGraph))
36	7, 34, 35	sylanbrc 594	1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ ComplUSGraph)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141   ∉ wnel 3062  ∀wral 3077  {crab 3414  Vcvv 3453   ∖ cdif 3901  {csn 4588  ⟨cop 4594   I cid 5555   ↾ cres 5663  ‘cfv 6536  Vtxcvtx 29312  Edgcedg 29363  UPGraphcupgr 29396  USGraphcusgr 29465  UnivVtxcuvtx 29701  ComplGraphccplgr 29725  ComplUSGraphccusgr 29726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-dju 9886  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-n0 12504  df-xnn0 12577  df-z 12591  df-uz 12862  df-fz 13535  df-hash 14367  df-vtx 29314  df-iedg 29315  df-edg 29364  df-uhgr 29374  df-upgr 29398  df-umgr 29399  df-uspgr 29466  df-usgr 29467  df-nbgr 29649  df-uvtx 29702  df-cplgr 29727  df-cusgr 29728

This theorem is referenced by:  cusgrsize  29770